Alice in Quantumland: An Allegory of Quantum Physics

by Robert Gilmore

  He stopped and looked around with a satisfied air. "Thank you for an interesting talk," said the Master, "that was very, very interesting. Does anyone have any questions?"

  Alice discovered that she did. Perhaps the school atmosphere was affecting her after all. She put her hand up. "Yes," said the Master, pointing to her, "what is the question that you would like to ask?"

  "There is one thing that I do not understand," said Alice. (This was not strictly true, as there were many things that she did not understand, and the number was becoming larger at a most alarming rate, but there was one particular thing about which she wished to ask a question.) "You say the world is customarily in this strange mixture of different states, but it reduces to one unique condition when you, as a conscious mind, happen to look at it. I suppose that any person is able to make something become real in this way, so what happens about other people's minds?"

  "We do not believe that We understand what you mean," replied the Emperor crushingly, but the Master cut in at this point.

  "Perhaps I might enlarge on the young lady's question. We were talking earlier about electrons passing through two slits. Suppose I were to take a photograph which would show an electron in the act of passing through either one slit or the other. If I follow what you say, you would maintain that, as the photograph might show that the electron was in either slit, it would have to show that it was in both. The photographic plate has no conscious mind and would be unable to reduce the wave function, so a superposition of two different images would be present on the film. Now suppose that I were to make a number of copies of this photograph, without of course looking at any of them. Would you say that each print would now also have a mixture of different images on it, each corresponding to the different slits which the electron might have passed through?"

  "Yes," replied the Emperor cautiously. "We believe that that would be the case."

  "If that is so and if the prints were all posted to different people, then the first one to open his envelope and look at the picture would cause one image from the mixture to become the real one and all the others would vanish?" Again the Emperor agreed cautiously. "But in that case, the photographs which the other people received would then each have to reduce to the same image, even though they might be in different cities miles apart. We know from experience that copies of a photograph do indeed show the same thing as the original, and if it was the occasion when the first person looked at a copy which caused one possibility to become uniquely real, presumably this act affected all the other copies, as they must subsequently agree with the first one. So one person who looked at a copy in one city would make all the other copies in other cities all over the world suddenly change to show the same thing. It would turn into a peculiar sort of race, with the first person to open the envelope fixing the images on all the other people's prints before they opened them. I think that was what the young lady meant," he finished.

  "Naturally such a consideration would not present any problem in Our case," responded the Emperor, "since no one would presume to look at such a photograph before We had examined it first. However, We see that such a situation might arise among people of the lower orders, and in that case the situation would indeed be as you describe."

  Alice was so startled at having this apparently ridiculous argument accepted that she did not notice the Emperor return to his seat and the little mermaid come up. The mermaid was unable to stand in front of the class, as she did not have any feet, so she sat on the Master's table, swinging her tail in front of her. Alice's attention returned to the proceedings as the mermaid began to speak.

  The Little Mermaid’s Theory (Many Worlds)

  "As you know," she began in a liquid, musical voice, "I am a creature of two worlds. I live in the sea and am equally at home upon the land. But this is as nothing compared with the number of worlds which we all inhabit, for we are all citizens of many worlds-many, many worlds.

  "The previous speaker told us that the quantum rules apply to the whole world, apart from the minds of the people who live in it. I tell you that they apply to the whole world, to everything. There is no limit to the idea of the superposition of states. When an observer looks at a superposition of quantum states you would expect him or her to see all of the effects that are appropriate to the selection of states present. This is what does happen; one observer does see all the results, or rather the observer also is in a superposition of different states, and each state of the observer has seen the result that goes with one of the states, in the original mixture. Each state is simply extended to include the observer in the act of seeing that particular state.

  "This is not the way that it seems to us, but that is because the different states of the observer are not aware of one another. When an electron passes through a screen with two slits in it, then it might pass through to the left or to the right. What you observe to happen is pure chance. You might see that the electron has gone to the left, but there will be another you that will have seen the electron go to the right. At the point at which you observe the electron, you split into two versions of yourself, one to see each possible result. If these two versions never get together again, then each remains totally unaware of the other's existence. The world has split into two worlds with slightly different versions of you in them. Of course, as these different versions of you will then talk to other people, you need different versions of them also, so what you have is a splitting of the entire universe. In this case it would split into two, but for a more complex observation it would split into a larger number of versions."

  "But surely this would happen rather often," Alice could not help herself from saying, interrupting the flow of the mermaid's talk.

  "It always happens," replied the mermaid calmly. "Whenever you have a situation where a measurement could give different results, then all the possible results will be observed, and the world will split into the appropriate number of versions.

  "Mostly the split worlds would remain separate and would diverge without ever being aware of one another, but sometimes they will come together again at some point and give interference effects. It is the presence of these interference effects between the different states which shows that they can and do all exist together."

  The mermaid stopped speaking and sat there combing the myriad strands of her long hair as they fell, side by side but separate, down over her shoulders.

  "It must mean there are an awful lot of universes. There would have to be as many as there are grains of sand on all the beaches on Earth," Alice protested.

  "Oh, there would be far more than that. Far more!" replied the mermaid dismissively. "Far, far more," she went on dreamily. "Far, far, far. . .."

  "That theory," cut in the Master, "has the advantage of being rather economical with assumptions, but it is very extravagant with universes!" He went on to ask for the next speaker. This was the Ugly Duckling, who had to stand on top of the Master's table so that he could be more clearly seen.

  The Ugly Duckling’s Theory (It Is All Too Complicated)

  The Duckling began his speech, and Alice observed that, as well as being very ugly, he appeared to be very cross as well. His speech was so full of quacks and spluttering that she was hard put to make out what he said. As far as she could tell he was saying that the superposition of different states only worked for rather small systems, with just a few electrons or atoms. He said that you need only argue that systems were often in mixtures of states because interference happened, since a single, unique, state would have nothing with which it could interfere.

  He further argued that you do not actually know that interference does happen for objects which contain many particles. People know that interference and hence the superposition of states can occur for groups of a few particles, so they think the same must still be true for complicated things, like ducklings. He would be quacked if he believed that.

  A duckling contains a lot of quacking atoms, he went on, and before any superimposed states ca
n interfere, all the atoms in each separate state must combine exactly with the appropriate atom in the other states. There are so many atoms that this is not quacking likely. Any effects would average out, and you could not see any net result. So how, he asked, can you be so quacking sure that ducklings are ever in a superposition of states? Answer me that if you are so quacked clever. All this superposition of states is fine and quacking for a few particles at a time, but it stops well short of ducklings.

  He went on to say that he quacking well knew when he saw something and when he quacking didn't. He knew that he was not in any quacking superposition of states, he was in only one, worse luck. So when he changed, he continued forcefully, he really changed from one definite state to another. The change was irreversible and there was no question of going back to combine with other states. Nothing was going to quacking interfere with him he concluded. At this point his quacking became so extreme that Alice could not follow him at all and was not really surprised when he became so angry that he fell off the table, out of her sight.

  There was a pause and a moment of silence. This ended as a long graceful neck rose from behind the desk, followed by a snowy white feathered body. It was a swan.

  "How beautiful!" exclaimed Alice. "May I stroke you?"

  The swan hissed at her furiously and clapped his wings in a threatening manner. Alice decided that, though his change was certainly irreversible, it did not appear to have changed his temper very much.

  At this point there was a disturbance at the back of the classroom, and Alice heard a voice shouting "Stop this charade, you are all wrong!" She looked across and saw a tall figure striding angrily down the space between the desks. It was the Classical Mechanic. His progress was considerable hampered by the fact that he was carrying a pinball machine, much as Alice had previously seen in cafes. (They might more often be found in bars, but of course Alice was too young to have seen them there.)

  The Classical Mechanic’s Theory (Wheels Within Wheels)

  The Classsical Mechanic marched to the front of the room and set his machine down by the Master's table. It was labeled "Electron Interceptor" and had the form of a sloping table, with two slits at the top through which the particles would be fired and a row of pockets along the bottom which were alternately marked "Win" and "No Win." The surface of the table, though brightly painted, appeared strangely free of the usual selection of obstacles and flippers which Alice had previously seen on pinball machines.

  "You are all deceiving yourselves," the Classical Mechanic announced firmly. "I have looked carefully at this device, which is basically a normal two-slit electron interference setup, and I believe I see what is really going on."

  Alice could see that, apart from its garish decoration, it was indeed a smaller version of the experiment which she had been shown in the Mechanics' gedanken room. The Classical Mechanic quickly demonstrated its operation by firing a stream of electrons from the two slits. At least Alice presumed that they must have come through those slits as they were the only ones present, although she was not able to see clearly where the electrons actually were until their arrival registered along the bottom of the table. As she had by now come to expect, the electrons clustered in a series of heaps, with gaps between the heaps where very few were detected. Alice was intrigued to see that these gaps in the interference pattern corresponded closely with the pockets marked Win.

  "You see that interference is produced and you would argue that this shows the electrons have somehow each come through both slits, so that the combination of the amplitudes for the two slits is producing the interference pattern we see. I tell you now that the electrons are in fact each going through just one slit, in a perfectly sensible way. The interference is due to hidden variables!"

  Alice found it very hard to follow exactly what happened at that point. The best she could say afterward was that the Classical Mechanic seemed to pull from the pinball table a dust cover which had not apparently been there before. However it had happened, she now saw that the surface of the table was covered with a pattern of deep ridges and grooves, leading away from the two slits. "Behold, hidden variables!" cried the Mechanic.

  "They are not very well hidden," remarked Alice, looking critically at the complicated surface now revealed.

  "My contention," began the Classical Mechanic, pointedly ignoring Alice's remark, "is that electrons and other particles behave in a perfectly rational and indeed classical fashion, very much like the particles to which I am accustomed in ClassicWorld. The only difference is that here, as well as the normal forces which act upon particles, they are also affected by a special quantum force, or pilot wave. This causes the strange effects which you interpret as due to interference. In my demonstration with the electron pinball here each electron really does enter by one slit or the other. It then moves over the table in a respectable and predictable fashion. Any randomness in the setup comes from the different directions and speeds which the electrons happen to have initially. When the electrons cross over the dips that you see here in the quantum potential, then the quantum force will deflect them, like a bicycle wheel caught in a trolley rail, so that most of the electrons end up in clumps. This gives your so-called interference effects."

  "Well now," said the Master, "that is certainly a very interesting theory-very, very interesting indeed. However, if you do not mind my saying so, you seem to have removed those difficulties you had with the electron's behavior at the expense of some very peculiar behavior for your quantum potential.

  "Because your quantum force has to produce the effects which we say are due to interference, it must be affected by things that happen in quite different places. If a third slit were to open in your table, then the quantum forces on the particles would change, even if none of the particles had gone through that hole. It must do this because the interference for three holes is different from that for two, and your force has to reproduce all those interference effects that we know to occur. Further your quantum potential, or network of quantum forces, must be very complicated indeed. In this theory you have nothing like the reduction of the wave functions which occurs in the normal quantum theory, so your potential must be affected by all the possibilities of everything which might have happened―ever. It is like the Many Worlds theory in that way. In your theory you say that what is observed will depend on how the particles happened to be traveling when they were affected by your pilot wave, but the pilot wave itself will retain information from all the possible things which might have happened and there is no way of removing it. Your wave would have to be incredibly complicated, like the sum of all the worlds in the Many Worlds theory, even though most of it may not affect any particles for most of the time.

  "The pilot wave in your theory affects what the particles do, but the way that the individual particles actually move has no effect on the wave. This depends only on what particles might have done. There is no symmetry of action and reaction between the particles and the pilot wave. As a Classical Mechanic this must be a worry to you. You would not want to contradict Newton's Law that action and reaction are always equal, would you now?"

  At this point the Quantum Mechanic, who had followed the Classical Mechanic into the room but remained quietly in the background, came forward and took his colleague by the arm. "Come along," he said. "You surely do not want to get involved in a charge of Classical Heresy by denouncing Newton's Laws. All this academic discussion of what electrons may or may not actually be doing is not for the likes of us. We are Mechanics. As a Mechanic, my main concern is that the Quantum laws do work and work well. When I calculate an amplitude for some process, this tells me what is likely to happen. It gives me the probabilities of different results and it does it accurately and reliably. It is not my job to worry about what the electrons are doing when I do not look at them, as long as I can tell what they are likely to be doing when I do look. That is what people pay me to do."

  He led his subdued colleague quietly to one side, then turned to Alic
e and asked, "Have you learned as much as you want to know about observers and measurements?"

  "Well," began Alice, "to tell you the truth I feel more confused than I was before I came here."

  "Right," interrupted the Quantum Mechanic emphatically. "I thought as much. You have learned quite as much as you want to. Come along with me now and see some of the results of quantum theory. Let me show you some of the features of Quantumland."

  Notes

  1. The "measurement problem" is that the selection of one single possibility and the reduction of all the other amplitudes is quite unlike other quantum behavior, and it is not obvious how it can occur. The problem is stated most simply in the form: How can you ever actually measure anything? The conventional view of quantum mechanics is that, when there are several possibilities, there will be present an amplitude for each one, and the overall amplitude for the system is the sum, or superposition, of all of them together. For example, if there are several slits through which a particle might pass, then the overall amplitude for the system contains an amplitude for each slit, and you can have interference between the individual amplitudes. If the system is left to itself, then the amplitudes will change in a smooth and predictable way. When you make a measurement on a system which has a sum of amplitudes corresponding to different possible values of the quantity measured, then the theory says that you will, with some probability, observe one or another of these values. Immediately after the measurement the value is a known quantity (because you have just measured it), so the sum of eigenstates (see box on p. 76) reduces to a single one, the one for the actual value you have just measured.